Simulation Results

In this section, a number of simulations are performed on time-varying vehicle model. The performance of the LPV/H∞ controller is compared with the LTI/H∞ in terms of

minimization of the yaw rate error e_ψ˙, deviation of the slip side angle and yaw rate, and the

generated yaw moment control input. It is noted that the corrective steering moment has a small magnitude which is only used to increase the handling performance over the entire operational envelope of the system. The commanded steering that is generated by a driver

19_{Note that we convert cornering stiffness values fromN/deg in Figure 3.11 to N/rd. Assume that the} dry road profile hasµ = 1 that corresponds to approximately 1050 N/deg, the conversion gives the number of 60000N/rd in our design.

or a controller initiates the lane change operation. The reference model responses are the reference tracking of the lane change trajectory, vehicle side slip angle, and the yaw rate that is to be followed by the designed LPV/H∞ controller in existence of the parameter

variation in the model. Therefore, the yaw rate ˙ψdis a function of the commanded steering.

Then, the goal is to define a suitable feedback control law that achieves good tracking and disturbance performance over the entire operating range. The proposed LPV/H∞ yaw

controller in Figure 3.12 is a single input multi output controller, which responds to the e_ψ˙, outputs the yaw moment and the corrective steering input. The scheduling parameters

Cf = Cr ∈ [30000, 60000]20. This range is defined according to Figure 3.11 and the

lower bound in ρ denotes a wet road profile (due to rain), whereas the upper bound of ρ presents a dry road profile. And the rates of change in parameter variation is assumed to be ˙Cf = ˙Cr ∈ [−20, 20] for computational purposes. The LMIs in Theorem 1 are

solved by the LMI Control Toolbox in Matlab R_{, the performance level L}

2 norm of the

LTI design is γ = 0.4243 while the LPV design is γ = 0.7892. This result ensures that the maximum attenuation of the disturbance vectors over the regulated outputs over the entire parameter range is less than γ i.e., kTzwki,2 < γ. The closed loop simulation is

performed in Simulink R

software where the LPV controllers are implemented using s- Functions. The duration of the LPV/H∞ on a computer equipped with Core i7 3.2 GHz

CPU is about 30 minutes. Furthermore, the LTI controller (fixed) is designed such that the variations of the road profile are kept constant. Then, the simulation is performed by assuming the cornering stiffness changes in the plant model.

It is important to state that a steering signal either from a driver or a controller26 to the reference model leads to a successful lane change maneuver with an optimal yaw rate and

20_{We did not consider a snowy road profile in highway driving where the cornering stiffness values would} be lower than that of the rainy road.

26_{Notice that the steering profile in Figure 3.16, activated at 5 seconds of the simulation, is produced as} a function of lane change maneuver signal and disturbance wind load [120].

Algorithm 2: Computation of controller matrices

Input: A(ρ), B1(ρ), B2(ρ), C1(ρ), D11(ρ), D12(ρ), C2(ρ)

D21(ρ), D22(ρ)

Output: Ac(ρ), Bc(ρ), Cc(ρ), Dc(ρ)

for ˙ρ ∈ [ ˙ρmin, ˙ρmax] do

for ρ ∈ [ρmin, ρmax] do

Form open loop nonlinear plant matrices (equation 3.26);

Solve the optimization problem with LMIs constraints at each grid points with equations (3.27) and (3.28);

end end

Compute controller matrices with equation (3.29) and equation (3.30).

0 1 2 3 4 5 6 7 8 9 10 3 3.5 4 4.5 5 5.5 6 6.5x 10

4 Cornering stiffness change in road profile during lane change

Time(sec)

Cornering stiffness (N/rd)

Change from dry road profile to wet road profile Cornering stiffness decreases

Change from wet road profile to dry road profile Cornering stiffness increases

Figure 3.14: Tire cornering stiffness change in road profile21 _{2017 IEEE. Reprinted,c}

with permission, from Serdar Coskun and Reza Langari, Enhanced vehicle handling per- formance for an emergency lane changing controller in highway driving, June 2017.

0 5 10 15 20 25 30 35 40 45 50 0 500 1000 1500 2000 2500 3000 Disturbance Time(sec) Wind Load (N) Disturbance Profile

Figure 3.15: Wind disturbance22 _{2017 IEEE. Reprinted, with permission, from Serdarc}

Coskun and Reza Langari, Enhanced vehicle handling performance for an emergency lane changing controller in highway driving, June 2017.

0 5 10 15 20 25 30 35 40 −1.5 −1 −0.5 0 0.5 1 1.5 Commanded steering Time(sec) Steering angle(deg)

Figure 3.16: Commanded steering24 _{2017 IEEE. Reprinted, with permission, from Ser-c}

dar Coskun and Reza Langari, Enhanced vehicle handling performance for an emergency lane changing controller in highway driving, June 2017.

Table 3.2: Vehicle parameters 23 _{2017 IEEE. Reprinted, with permission, from Serdarc}

Coskun and Reza Langari, Enhanced vehicle handling performance for an emergency lane changing controller in highway driving, June 2017.

Symbol Value Unit Description

m 1500 kg Vehicle mass

mz 350 kg Vehicle rear mass

J 2500 kgm2 _{Yaw inertia}

lf 1.1 m Distance of front axle to COG

lr 1.5 m Distance of rear axle to COG

Cf [30000,60000] N/rd Varying front cornering stiffness

Cr [30000,60000] N/rd Varying rear cornering stiffness

tf = tr 1.4 m Front and rear axle length

Vx 100 km/h Vehicle longitudinal velocity

WR 3 m Width of the road

Wv 1.847 m Wind load acting distance to COG

slip performance in Figure 3.17. Notice that the wind disturbance effect is only employed in reference model so that the outputs present the effect of wind load. Then this yaw rate is a desired yaw rate to be followed by the LPV and LTI controllers that react to the nonlinear model. We consider a scenario where there is a sharp decrease and increase in cornering stiffness values (sudden change from a dry road to a wet road profile and vice versa) during a lane change in Figure 3.14. Figures 3.17, 3.18, 3.19 show the regulated output responses for both LPV and LTI cases for this scenario. It is observed that the LPV presents better tracking performance than that of the LTI controller in both Figure 3.17 and Figure 3.18. The deviations from the nominal value are satisfactory. Notice that the uncontrolled yaw rate performance is the worst in terms of reference following and disturbance rejection27. Moreover, the generated yaw moment control input in LPV design, as expected (due to the higher performance index value γ), is slightly bigger than that of the LTI design in Fig

27_{The magnitude of the yaw rate almost two times bigger than that of the LPV controller case. We reach} to our goal to enhance the yaw rate for the nonlinear vehicle model.

0 5 10 15 20 25 30 35 40 −15 −10 −5 0 5 10 15

Yaw rate outputs

Time(sec)

Yaw rate (deg/sec)

Desired LTI LPV Uncontrolled

Figure 3.17: Yaw rate outputs25 2017 IEEE. Reprinted, with permission, from Serdarc Coskun and Reza Langari, Enhanced vehicle handling performance for an emergency lane changing controller in highway driving, June 2017.

3.19. Note that the exertion of the yaw moment to the brake forces by taking into account actuator dynamics is a possible future direction in our research. To better analyze the handling performance over the varying cornering stiffnesses, Figure 3.20 is plotted to show the relation between the front steering angle and the yaw rate. This result simply means that deterioration of yaw rate of the system is affected to parameter changes with respect to the steering angle. It is shown that the LPV design better follows the desired value than the LTI case. It is interpreted that if the commanded steering is applied to the nonlinear model with no control, the generated yaw rate results in poor handling with respect to the degree steering angle. This result certainly assures that a safe lane change, initiated by the steering angle with an improved rate tracking even when the road conditions are varying. We can conclude that both the LTI and LPV enhance the handling performance with respect to the parameter changes. But, the LPV design provides better results in each case.

0 5 10 15 20 25 30 35 40 −1.5 −1 −0.5 0 0.5 1 1.5

Side slip outputs

Time(sec)

Side slip angle (deg)

Desired LTI LPV

Figure 3.18: Slip side angle outputs 28 2017 IEEE. Reprinted, with permission, fromc Serdar Coskun and Reza Langari, Enhanced vehicle handling performance for an emer- gency lane changing controller in highway driving, June 2017.

0 5 10 15 20 25 30 35 40 −600 −400 −200 0 200 400 600 Yaw Moment Time(sec) Yaw Moment(Nm) LTI LPV

Figure 3.19: Yaw moments 29 2017 IEEE. Reprinted, with permission, from Serdarc Coskun and Reza Langari, Enhanced vehicle handling performance for an emergency lane changing controller in highway driving, June 2017.